Polyadic Sets and Homomorphism Counting

TL;DR

This paper introduces a categorical framework using polyadic sets to analyze homomorphism counts, extending classical graph isomorphism results to infinite structures with new counting theorems.

Contribution

It develops a novel categorical approach based on polyadic sets for homomorphism counting, applicable to both finite and infinite structures.

Findings

Categorical framework for homomorphism counting using polyadic sets

Extension of Lovasz's result to infinite structures like trees and profinite algebras

New homomorphism counting theorems for various classes of structures

Abstract

A classical result due to Lovasz (1967) shows that the isomorphism type of a graph is determined by homomorphism counts. That is, graphs G and H are isomorphic whenever the number of homomorphisms from K to G is the same as the number of homomorphisms from K to H for all graphs K. Variants of this result, for various classes of finite structures, have been exploited in a wide range of research fields, including graph theory and finite model theory. We provide a categorical approach to homomorphism counting based on the concept of polyadic (finite) set. The latter is a special case of the notion of polyadic space introduced by Joyal (1971) and related, via duality, to Boolean hyperdoctrines in categorical logic. We also obtain new homomorphism counting results applicable to a number of infinite structures, such as finitely branching trees and profinite algebras.